How to make cloth 'catch' a falling object, slowing its fall?
Here is an example, based on what you said, if I got it right:


subdivide a bit a cube, rise it on Z, set rigid body "active", a bit of bounciness, "mesh" collision shape
set a ground plane, set a group with the 4 corner vertices, set plane to cloth, with pin to the above 4 vertices group, set also rigid body "passive", with a bit of bounciness, set collision shape to "mesh"


it should behave like this:



here is the example file



Hth.

edit

As for comments, I noticed you asked something slightly more natural. Rigid bodies made the cloth stop the falling cube, but it seemed as their interaction was poor. 

Although you didn't post any image, animation or example file, I hope now to understand what you wish, and have done another setup, but it does not use rigid bodies, it uses soft bodies:

Here is the result:



I've done it like this:

Plane has


cloth enabled, pinned to the 4 corner vertices group
collision enabled (default settings)
soft body enabled with "default goal strength" .98


Falling cube has also soft body enabled, with


soft body goal > disabled
soft body edges > stiff quads, shear 1
soft body edges > springs pull 0.99


And, here is another example file



edit 2:

since in comments you (finally!) unveiled your ultimate desire: 

"I specifically wanted to achieve the effect of a (soccer) football being caught by a goal net."

I tried similar setups and... found that's is really hard, imho. I had some partial (weird) results using mostly softbodies instead of cloth, and you see them below, but although I'm not satisfied (all of them are really acting weirdly) I show them here below for completeness's sake. To partially justify my failures, I found this on this manual page: Deflection for softbody objects is difficult, they often penetrate the colliding objects.

Below you see I used a "force" simulation (with a abrief strong impulse behind the ball) to "kick" the ball into the net, this also could help or inspire you. Last, I would create a new question more specific, like "how can I realistically simulate a soccer ball caught in a goal net?"...

So, here are some results I got until now: I can post also .blend files if you need to experiment: it was fun, in the end, but I wish I found a very good end compelte solution...







late edit

here is the .blend file relative to the last example